Theorem canthwdom 9537

Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 9114, equivalent to canth 7362). (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
canthwdom	⊢ ¬ 𝒫 𝐴 ≼* 𝐴

Proof of Theorem canthwdom

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0elpw 5324	. . . . 5 ⊢ ∅ ∈ 𝒫 𝐴
2		ne0i 4302	. . . . 5 ⊢ (∅ ∈ 𝒫 𝐴 → 𝒫 𝐴 ≠ ∅)
3	1, 2	mp1i 14	. . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝒫 𝐴 ≠ ∅)
4		brwdomn0 9527	. . . 4 ⊢ (𝒫 𝐴 ≠ ∅ → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴))
5	3, 4	syl 18	. . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴))
6	5	ibi 270	. 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)
7		relwdom 9524	. . . . 5 ⊢ Rel ≼*
8	7	brrelex2i 5716	. . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝐴 ∈ V)
9		foeq2 6787	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝑥))
10		pweq 4578	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
11		foeq3 6788	. . . . . . . 8 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴))
12	10, 11	syl 18	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴))
13	9, 12	bitrd 282	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴))
14	13	notbid 321	. . . . 5 ⊢ (𝑥 = 𝐴 → (¬ 𝑓:𝑥–onto→𝒫 𝑥 ↔ ¬ 𝑓:𝐴–onto→𝒫 𝐴))
15		vex 3467	. . . . . 6 ⊢ 𝑥 ∈ V
16	15	canth 7362	. . . . 5 ⊢ ¬ 𝑓:𝑥–onto→𝒫 𝑥
17	14, 16	vtoclg 3531	. . . 4 ⊢ (𝐴 ∈ V → ¬ 𝑓:𝐴–onto→𝒫 𝐴)
18	8, 17	syl 18	. . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ 𝑓:𝐴–onto→𝒫 𝐴)
19	18	nexdv 1963	. 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)
20	6, 19	pm2.65i 196	1 ⊢ ¬ 𝒫 𝐴 ≼* 𝐴

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463  ∅c0 4294  𝒫 cpw 4564   class class class wbr 5110  –onto→wfo 6531   ≼^* cwdom 9522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-wdom 9523

This theorem is referenced by:  pwdjudom  10194